Fault detection in electric power delivery systems using underreach, directional, and traveling wave elements

ABSTRACT

The present disclosure pertains to systems and methods for detecting faults in an electric power delivery system. In one embodiment, a system may include a data acquisition subsystem configured to receive a plurality of representations of electrical conditions. The system may also include a traveling wave differential subsystem configured to determine an operating quantity based on the plurality of representations of electrical conditions. The traveling wave differential subsystem may also determine a restraint quantity based on the plurality of representations of electrical conditions. The traveling wave differential subsystem may detect a traveling wave generated by the fault based on the plurality of representations. A fault detector subsystem may be configured to declare a fault based on a comparison of the operating quantity and the restraint quantity. A protective action subsystem may implement a protective action based on the declaration of the fault.

RELATED APPLICATION

The present application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Patent Application No. 62/051,193, filed Sep. 16, 2014, andtitled “Fault Detection in Electric Power Delivery Systems UsingUnderreach, Directional, and Traveling Wave Elements,” which isincorporated herein by reference in its entirety.

TECHNICAL FIELD

This disclosure relates to detecting and locating faults in electricpower delivery systems. More particularly, this disclosure relates tousing time domain elements and analysis to determine fault location inelectric power delivery systems. In various embodiments, systems andmethods consistent with the present disclosure may utilizeunderreach/overreach, directional, and/or traveling wave elements.

BRIEF DESCRIPTION OF THE DRAWINGS

Non-limiting and non-exhaustive embodiments of the disclosure aredescribed, including various embodiments of the disclosure withreference to the figures, in which:

FIG. 1 illustrates a block diagram of a system for detecting a travelingwave and calculating a location of a fault using the detected travelingwave consistent with certain embodiments of the present disclosure.

FIG. 2A illustrates a lattice diagram showing incident and reflectedtraveling waves over a relative time scale created by a fault event on a300 mile (482.8 km) long transmission line consistent with certainembodiments of the present disclosure.

FIG. 2B illustrates the incident and reflected traveling waves as afunction of current over time from the fault illustrated in FIG. 2Aconsistent with certain embodiments of the present disclosure.

FIG. 2C illustrates a lattice diagram showing the incident and reflectedtraveling waves at a remote terminal and a local terminal from a faultevent on a 400 km long transmission line consistent with certainembodiments of the present disclosure.

FIG. 3 illustrates a conceptual representation of a faulted electricalnetwork as an equivalent of the sum of a pre-fault network and a faultnetwork consistent with certain embodiments of the present disclosure

FIG. 4A illustrates an equivalent single-phase network with a localTerminal S and a remote Terminal R, with a fault on a line betweenterminals S and R consistent with certain embodiments of the presentdisclosure.

FIG. 4B illustrates another equivalent single-phase network that hasbeen simplified for analysis of incremental quantities consistent withcertain embodiments of the present disclosure.

FIG. 5A illustrates a plot over time of the incremental voltage andincremental replica current for a forward fault consistent with certainembodiments of the present disclosure.

FIG. 5B illustrates a plot over time of the incremental replica voltageand incremental replica current for a reverse fault consistent withcertain embodiments of the present disclosure.

FIG. 6 illustrates a plot over time of a direction element operation fora forward fault consistent with certain embodiments of the presentdisclosure.

FIG. 7A illustrates a plot over time of an incremental voltage, for anin-zone fault occurring near the voltage peak and located atapproximately 60% of the set reach point of a directional elementconsistent with one embodiment of the present disclosure.

FIG. 7B illustrates a plot over time of an incremental replica currentunder the same conditions illustrated in FIG. 7A consistent with oneembodiment of the present disclosure.

FIG. 7C illustrate a plot over time of an operating signal under thesame conditions illustrated in FIG. 7A consistent with one embodiment ofthe present disclosure.

FIG. 8 illustrates an operating characteristic of an underreachingelement with directional supervision consistent with certain embodimentsof the present disclosure.

FIG. 9 illustrates an operating characteristic of an overreachingnondirectional element consistent with certain embodiments of thepresent disclosure.

FIG. 10 illustrates a flowchart of one embodiment of a method fordetermining a direction to a fault using time-domain quantitiesconsistent with certain embodiments of the present disclosure.

FIG. 11 illustrates a flow diagram of one embodiment of a method fordetermining whether a fault is within a zone of protection consistentwith the present disclosure.

FIG. 12 illustrates one embodiment of a method for determining adirection to a fault and for determining whether the fault is within azone of protection consistent with certain embodiments of the presentdisclosure.

FIG. 13 illustrates a block diagram of a system configured to use of acorrelation technique for determining a fault location consistent withcertain embodiments of the present disclosure.

FIG. 14A illustrates TWs launched by a fault and reflected at locationsU, S, and R consistent with certain embodiments of the presentdisclosure.

FIG. 14B illustrates a plot over time of traveling wave currentsreceived at Terminal S and Terminal R in FIG. 14A during an externalfault consistent with certain embodiments of the present disclosure.

FIG. 14C illustrates a plot over time of traveling wave currentsreceived at Terminal S and Terminal R in FIG. 14A during an externalfault consistent with certain embodiments of the present disclosure.

FIG. 14D illustrates a plot of traveling wave currents received atTerminal S and Terminal R in FIG. 14A during an internal faultconsistent with certain embodiments of the present disclosure.

FIG. 15 illustrates a flow diagram of a method for determining internalfault conditions using a traveling wave differential module consistentwith certain embodiments of the present disclosure.

FIG. 16 illustrates a functional block diagram of a system for detectingfaults and estimating a fault location using traveling waves consistentwith certain embodiments of the present disclosure.

DETAILED DESCRIPTION

Faster transmission line protection improves power system stability. Iffaults are not cleared before the critical fault clearing time, thesystem may lose transient stability and possibly suffer a black out. Inaddition, faster fault clearing increases the amount of power that canbe transferred. Faster protection also enhances public and utilitypersonnel safety, limits equipment wear, improves power quality, andreduces property damage.

Most protection principles are based on the fundamental frequencycomponents of voltages and currents. Accurate measurement of asinusoidal quantity typically takes a cycle. To increase the speed ofprotection actions, an analysis of transient components may beundertaken in connection with various embodiments of the presentdisclosure. Further, information relating to electrical conditions maybe communicated among devices to provide end-to-end transmission lineprotection.

Primary protective relaying systems typically operate in one toone-and-a-half cycles, and the breakers interrupt current inone-and-a-half to three cycles, so faults are typically cleared in threeto four cycles. Sometimes the relaying system operates faster. Forexample, sensitive instantaneous overcurrent elements can be used forswitch-onto-fault events, and may have an operation time as low asone-quarter of a cycle. Traditional frequency domain techniques obtainedby extracting fundamental frequency components (phasors) may be appliedto identify a fault after transient signals fade. The filteringnecessary for phasor measurement results in operating times of about onepower cycle, with the best-case times approaching half a cycle forclose-in high-current faults.

However, for purposes of determining stability limits for planningpurposes, it is most appropriate to utilize conservative protectionoperating times. If a breaker fails to trip, breaker failure schemestake over, and fault clearing is delayed until the slowest backupbreaker operates, which may be around 10 to 12 cycles. Iftime-coordinated remote backup protection is used instead of breakerfailure protection, the fault clearing time may be as high as a fewhundred milliseconds.

High-speed protection devices respond to high-frequency signalcomponents, which may be used to detect faults and to realize variousadvantages. For example, certain nontraditional energy, such as wind andsolar, are connected to the power system through a power electronicsinterface. As such, these sources typically have little or no inertia.Their control algorithms protect the converters for network faultconditions. As a result, these sources produce voltages and currentsthat challenge some protection principles developed for networks withsynchronous generators. In contrast, high-speed protection devicesconfigured to respond to high-frequency signal components are lessdependent on the sources and more dependent on the network itself. As aresult, such relays may be useful in applications near nontraditionalsources.

Various embodiments consistent with the present disclosure may analyzetraveling waves (TWs) to aid in the detection of faults. When a faultoccurs in an electric power system, traveling waves are launched fromthe fault and travel outward at a velocity near the speed of light. Thetraveling waves are reflected by buses and other discontinuitiesaccording to their corresponding characteristic impedances. In theinitial stage of the fault, the electric power system may behave like adistributed parameter network. Accordingly, the traveling waves may bedescribed by the propagation velocity, the reflection and transmissioncoefficients, and the line characteristic impedance. Using a travelingwave detection algorithm, a high-speed relay may be able to detect afault and initiate corrective action in less than 1 millisecondconsistent with certain embodiments of the present disclosure.

After a few roundtrip reflections, traveling waves from a faultrecombine into stationary waves, and the power system may beapproximated using a lumped parameter RLC network in a transient state.Given the speed of traveling waves, such a condition may be realizedvery shortly following the occurrence of a fault. TWs from a faultanywhere on a 100-mile line reach both ends within 600 microseconds.Various embodiments consistent with the present disclosure may analyzethe “lumped circuit theory” transient waveforms to detect a fault andinitiate corrective action within milliseconds consistent with certainembodiments of the present disclosure.

Various techniques may be used to simplify models utilized in certainembodiments. For example, certain embodiments may analyze incrementalquantities, which are signals that appear due to a fault and do notcontain load voltages or currents. Incremental quantities may simplifythe line and system representation by eliminating power sources andleaving the fault as the only “source” in the equivalent network. Inother words, the driving force of the transient is the fault, and thedriving force of the steady-state response is the set of systemfundamental frequency sources (e.g., generators).

Ultra-high-speed principles allow relays to identify events that arelocated within the protected zone but are not necessarily permanentfaults. Incipient cable failures or surge arrester conduction events maypresent detection challenges to existing feeder and bus relays,respectively. Similarly, the ultra-high-speed line protection needs toensure that an in-zone event is a legitimate fault.

The embodiments of the disclosure will be best understood by referenceto the drawings, wherein like parts are designated by like numeralsthroughout. It will be readily understood that the components of thedisclosed embodiments, as generally described and illustrated in thefigures herein, could be arranged and designed in a wide variety ofdifferent configurations. Thus, the following detailed description ofthe embodiments of the systems and methods of the disclosure is notintended to limit the scope of the disclosure, as claimed, but is merelyrepresentative of possible embodiments of the disclosure. In addition,the steps of a method do not necessarily need to be executed in anyspecific order, or even sequentially, nor need the steps be executedonly once, unless otherwise specified.

In some cases, well-known features, structures or operations are notshown or described in detail. Furthermore, the described features,structures, or operations may be combined in any suitable manner in oneor more embodiments. It will also be readily understood that thecomponents of the embodiments as generally described and illustrated inthe figures herein could be arranged and designed in a wide variety ofdifferent configurations.

Several aspects of the embodiments described may be illustrated assoftware modules or components. In other embodiments,hardware-implemented embodiments may be used. Such embodiments mayutilize, among other technologies, field-programmable gate arrays. Asused herein, a software module or component may include any type ofcomputer instruction or computer executable code located within a memorydevice and/or transmitted as electronic signals over a system bus orwired or wireless network. A software module or component may, forinstance, comprise one or more physical or logical blocks of computerinstructions, which may be organized as a routine, program, object,component, data structure, etc., that performs one or more tasks orimplements particular abstract data types.

In certain embodiments, a particular software module or component maycomprise disparate instructions stored in different locations of amemory device, which together implement the described functionality ofthe module. Indeed, a module or component may comprise a singleinstruction or many instructions, and may be distributed over severaldifferent code segments, among different programs, and across severalmemory devices. Some embodiments may be practiced in a distributedcomputing environment where tasks are performed by a remote processingdevice linked through a communications network. In a distributedcomputing environment, software modules or components may be located inlocal and/or remote memory storage devices. In addition, data being tiedor rendered together in a database record may be resident in the samememory device, or across several memory devices, and may be linkedtogether in fields of a record in a database across a network.

Embodiments may be provided as a computer program product including amachine-readable medium having stored thereon instructions that may beused to program a computer (or other electronic device) to performprocesses described herein. The machine-readable medium may include, butis not limited to, hard drives, floppy diskettes, optical disks,CD-ROMs, DVD-ROMs, ROMs, RAMs, EPROMs, EEPROMs, magnetic or opticalcards, solid-state memory devices, or other types ofmedia/machine-readable medium suitable for storing electronicinstructions.

FIG. 1 illustrates a block diagram of a system 100 for detecting andcalculating a location of a fault using time-domain principles andelements further described herein. System 100 may include generation,transmission, distribution and/or similar systems. System 100 includes aconductor 106 such as a transmission line connecting two nodes, whichare illustrated as a local terminal 112 and a remote terminal 114. Localand remote terminals 112 and 114 may be buses in a transmission systemsupplied by generators 116 and 118, respectively. Although illustratedin single-line form for purposes of simplicity, system 100 may be amulti-phase system, such as a three-phase electric power deliverysystem.

System 100 is monitored by IEDs 102 and 104 at two locations of thesystem, although further IEDs may also be utilized to monitor furtherlocations of the system. As used herein, an IED (such as IEDs 102 and104) may refer to any microprocessor-based device that monitors,controls, automates, and/or protects monitored equipment within system100. Such devices may include, for example, remote terminal units,differential relays, distance relays, directional relays, feeder relays,overcurrent relays, voltage regulator controls, voltage relays, breakerfailure relays, generator relays, motor relays, automation controllers,bay controllers, meters, recloser controls, communications processors,computing platforms, programmable logic controllers (PLCs), programmableautomation controllers, input and output modules, and the like. The termIED may be used to describe an individual IED or a system comprisingmultiple IEDs. IEDs 102 and 104 may obtain electric power systeminformation using current transformers (CTs), potential transformers(PTs), Rogowski coils, voltage dividers and/or the like. IEDs 102, 104may be capable of using inputs from conventional instrument transformerssuch as CTs and PTs conventionally used in monitoring of electric powerdelivery. IEDs 102 and 104 may also receive common time information froma common time source 110.

Common time source 110 may be any time source capable of delivering acommon time signal to each of IEDs 102 and 104. Some examples of acommon time source include a Global Navigational Satellite System (GNSS)such as the Global Positioning System (GPS) delivering a time signalcorresponding with IRIG, a WWVB or WWV system, a network-based systemsuch as corresponding with IEEE 1588 precision time protocol, and/or thelike. According to one embodiment, common time source 110 may comprise asatellite-synchronized clock (e.g., Model No. SEL-2407, available fromSEL). Further, it should be noted that each IED 102, 104 may be incommunication with a separate clock, such as a satellite-synchronizedclock, with each clock providing each IED 102, 104 with a common timesignal. The common time signal may be derived from a GNSS system orother time signal.

A data communication channel 108 may allow IEDs 102 and 104 to exchangeinformation relating to, among other things, voltages, currents.time-domain fault detection and location. According to some embodiments,a time signal based on common time source 110 may be distributed toand/or between IEDs 102 and 104 using data communication channel 108.Data communication channel 108 may be embodied in a variety of media andmay utilize a variety of communication protocols. For example, datacommunication channel 108 may be embodied utilizing physical media, suchas coaxial cable, twisted pair, fiber optic, etc. Further, datacommunication channel 108 may utilize communication protocols such asEthernet, SONET, SDH, or the like, in order to communicate data.

In several embodiments herein, traveling waves on the electric powerdelivery system may be used to detect and calculate location of a fault.Two-end fault locating methods, which may be referred to herein as TypeD methods, may use a time difference between a traveling wave capturedat both terminals along with the line length and wave propagationvelocity to compute the fault location. Measurement devices at the lineterminals detect the traveling waves and time stamp the arrival of thewave using a common time reference (e.g., IRIG-B or IEEE 1588). Incertain embodiments, a distance to a fault location (m) is calculatedusing Eq. 1.

$\begin{matrix}{m = {\frac{1}{2}\lbrack {L + {( {t_{L} - t_{R}} ) \cdot v}} \rbrack}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$where: t_(L) is the front wave arrival time at the L Terminal,

-   -   t_(R) is the front wave arrival time at the R Terminal,    -   v is the wave propagation speed,    -   L is the line length.

Traditionally these solutions use a master station that accesses thewave arrival times and estimates the fault location. Recently, linerelays equipped with traveling wave fault locating function may exchangethe wave arrival times, calculate the fault location, and make the faultlocation available at the relay. One of the key benefits of using theType D method is its simplicity and immunity to reflections.

FIG. 2A illustrates a lattice diagram 200 showing incident and reflectedtraveling waves created by a fault consistent with certain embodimentsof the present disclosure. In the illustrated embodiment, a fault islocated 50 miles (80.5 km) from a first terminal on a 300 mile (482.8km) long line. The incident wave triggered by the fault reaches theTerminal L at time TL₅₀, and reaches the Terminal R at time TR₂₅₀. TheType D method may use the TL₅₀ and TR₂₅₀ to compute the fault locationwhile ignoring all the other waves. When desired, remaining wavearrivals can be used to improve the initial fault location estimate.

FIG. 2B illustrates the incident and reflected traveling waves as afunction of current over time 202 from the fault illustrated in FIG. 2Aconsistent with certain embodiments of the present disclosure. Asillustrated, the magnitude of the reflected traveling waves diminisheswith each reflection. Time alignment of data samples received at bothTerminal L and Terminal R allows for comparison of the incident andreflected waves from both terminals.

A single-end fault locating method, which is also referred to herein asa Type A fault locating method, uses the time difference between thefirst arrived traveling wave and a subsequent reflection from the faultor the remote terminal. The Type A method is not dependent on acommunication channel to the remote terminal. However, the challenge isto identify and select the appropriate reflection. The Type A method maybe useful, according to some embodiments, when the fault location iscomputed during reclosing events on a permanent fault when one of theterminals is open.

FIG. 2B illustrates the reflections from the fault at terminal L. Thepolarity, amplitude, and arrival time of the reflected waves can be usedto identify the reflected wave from the fault or the remote terminal andcalculate the fault location. At the L Terminal, the Type A method mayuse points labeled TL₅₀ and TL₁₅₀ in FIG. 2B to compute the faultlocation while ignoring other waves and reflections. In certainembodiments, a distance to a fault location (m) may be calculated usingthe Type A method using Equation 2.

$\begin{matrix}{m = {( \frac{t_{L\; 2} - t_{L\; 1}}{2} ) \cdot v}} & {{Eq}.\mspace{14mu} 2}\end{matrix}$where: t_(L2) is the arrival time of the first reflection from the faultat the L Terminal;

-   -   t_(L1) is the arrival time of the initial wave front from the        fault at the L Terminal; and    -   v is the wave propagation speed.

In various embodiments, the polarity of the traveling wave may be usedto determine the direction to the fault. Voltage and current polaritiesare opposite if the fault is in the forward direction. If the fault isin the reverse direction, the voltage and current traveling waves havesame polarity.

FIG. 2C illustrates a lattice diagram 204 showing the incident andreflected traveling waves at a remote terminal and a local terminal froma fault event on a 400 km long transmission line consistent with certainembodiments of the present disclosure. Assuming a 3×10⁸ m/s propagationvelocity, a fault located at 50 km on a 400 km line would result in atime lag between the initial front-wave and the first legitimatereflection from the fault that may be calculated using Eq. 3.

$\begin{matrix}{\frac{2 \times 50 \times 10^{3}}{3 \times 10^{8}} = {333\mspace{14mu}{µs}}} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

Further, knowing that the line is 400 km long, it is possible to obtaina delay time estimate for the first wave reflected from the remoteterminal. With respect to the instant of fault occurrence, the firstreflection from the remote terminal will be per Eq. 4.

$\begin{matrix}{\frac{( {{2*400} - 50} )*10^{3}}{3*10^{8}} = {2,500\mspace{14mu}{µs}}} & {{Eq}.\mspace{14mu} 4}\end{matrix}$As illustrated in FIG. 2C, a local relay generates measurement withrespect to the first arriving wave, which is 166.6 μs, because of the 50km distance between the local relay and the fault. The estimatedetermined using Eq. 4 may provide a window in which a reflected wavemay be expected after the initial front wave.

While the previous two-ended and single-ended traveling wave faultlocation methods provided a more accurate estimate of the location ofthe fault than was available using, for example, impedance-basedmethods, these methods were constrained due to communication systemlimitations and reliance on frequency-domain measurements. In thefrequency domain, measurements of electric power system voltage andcurrent require a full electric power system cycle to calculate withadequate accuracy. Thus, previous fault detection and locationalgorithms could not determine a location of a fault faster than oneelectric power system cycle, for most faults.

The time-domain electric power system fault detection and locationtechniques described herein do not require a complete electric powersystem cycle to calculate measurements of voltage or current.Conventional PTs and CTs may be used to provide signals corresponding tothe voltage and current of the electric power delivery system, which maybe used for fault detection and location calculations in less than oneelectric power system cycle.

FIG. 3 illustrates a conceptual representation of a faulted electricalnetwork 300 c as an equivalent of the sum of a pre-fault network 300 aand a fault network 300 b consistent with certain embodiments of thepresent disclosure. Thus, the faulted network 300 c may be solved byanalyzing separately the pre-fault network 300 a with the load portionof the currents and voltage and the faulted network 300 c to obtain thefault generated components of the currents and voltages. The actualsolution of currents and voltages at any point in the faulted networkare calculated as sums of the pre-fault and fault-generated components.

The pre-fault network 300 a may be in steady state. The fault network300 b has only one source (the Thévenin's source) and an equivalentimpedance. The Thévenin's source voltage is the voltage at the faultpoint in the pre-fault network. Before the fault, the fault network 300b is not energized and all of its voltages and currents are zero. When afault occurs, the fault network 300 b goes through a transient state andeventually settles into the fault steady state. The fault network 300 bcan be solved for the steady-state values or for the transient values.The incremental quantities are valid for both states. By their nature,the fault-generated signal components are not affected by load, but aredriven by a single source in the fault network located at the faultpoint. As a result, these quantities depend mostly on the networkparameters.

The fault signals in the fault network 300 b may be measured directly bymonitoring the instantaneous voltages and currents during the fault.Because the fault signals are sums of the pre-fault signals and thefault-generated signals, the fault-generated signals in fault network300 b are the difference between the fault signals in faulted network300 c and the pre-fault signals in pre-fault network 300 a. Thepre-fault signals in pre-fault network 300 a may be measured prior tothe fault and extrapolated beyond the fault. This extrapolation is validonly for a period of few tens to few hundred milliseconds because thepower system sources remain stationary in this short period of time.Therefore, a very simple method to derive incremental quantities in thetime-domain is illustrated in Equation 5:Δx _((t)) =x _((t)) −x _((t−p·T))  Eq. 5where:

Δx is the instantaneous incremental quantity;

x is the measured instantaneous value;

T is the power system cycle; and

p is a number of power cycles.

Equation 5 yields an incremental quantity (Δx) that lasts for p powercycles, after which the quantity expires because the historical valuesubtracted (x_((t−p·T))) slides into the fault period. The value of pmay be selected depending on the intended usage of the incrementalquantity. For example, if it is intended to use incremental quantitiesduring two power cycles, the p is selected as p>2, such as p=3.

The time-domain incremental quantities obtained from Equation 5 containall frequency components of the fault-generated signal. Thefundamental-frequency component does not include the effect of load.Depending on the usage of the incremental quantities, the signal fromEquation 5 may be further filtered to obtain the frequencies ofinterest. For example, fundamental-frequency band-pass filtering may beperformed, and Equation 5 applied in the frequency domain using phasorquantities.

FIG. 4A illustrates an equivalent single-phase network with a localTerminal S and a remote Terminal R, with a fault on a line betweenterminals S and R consistent with certain embodiments of the presentdisclosure. The fault network contains incremental voltages and currentsthat may be used for protection of the network as further describedherein. FIG. 4B illustrates another equivalent single-phase network thathas been simplified for analysis of incremental quantities consistentwith certain embodiments of the present disclosure. At the location S ofthe network, the incremental voltage and current are related by a simplevoltage drop equation across the Source S resistance and inductanceillustrated by Equation 6:

$\begin{matrix}{{\Delta\; v} = {- ( {{{R_{S} \cdot \Delta}\; i} + {{L_{S} \cdot \frac{\mathbb{d}\;}{\mathbb{d}t}}\Delta\; i}} )}} & {{Eq}.\mspace{14mu} 6}\end{matrix}$

Equation 6 may be scaled for the ease of use by multiplying and dividingeach side by the Source S impedance as shown in Equation 7:

$\begin{matrix}{{\Delta\; v} = {{- {Z_{S}}}( {{{\frac{R_{S}}{Z_{S}} \cdot \Delta}\; i} + {{\frac{L_{S}}{Z_{S}} \cdot \frac{\mathbb{d}\;}{\mathbb{d}t}}\Delta\; i}} )}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$

The current term in Equation 7 may be replaced by a new current termthat is the combination of the instantaneous incremental current and itsderivative as shown in Equations 8 and 9:

$\begin{matrix}{{\Delta\; i_{Z}} = {{{D_{0} \cdot \Delta}\; i} + {{D_{1} \cdot \frac{\mathbb{d}\;}{\mathbb{d}t}}\Delta\; i}}} & {{Eq}.\mspace{14mu} 8}\end{matrix}$where:

$\begin{matrix}{{D_{0} = \frac{R_{S}}{Z_{S}}},{D_{1} = \frac{L_{S}}{Z_{S}}}} & {{Eq}.\mspace{14mu} 9}\end{matrix}$

Equation 7 may then be re-written as Equation 10:Δv=−|Z _(S) |·Δi _(Z)  Eq. 10

Equation 10 is similar to the voltage-current expression for phasors inthe frequency domain, as shown in Equation 11:ΔV=−Z _(S) ·ΔI  Eq. 11

The current term Δi_(Z) is referred to herein as the “replica current”.The replica current may allow for the re-use—in the time domain—ofexpressions from the frequency domain such as those used with phasors.By selecting D₀ and D₁ coefficients a unity gain may be obtained betweenthe measured current and the replica current at the system fundamentalfrequency. The unity gain may be useful for setting an element usingtime-domain quantities as further described hereinafter.

For reverse faults, the Δv_(F) source of FIG. 4B may be placed behindthe S terminal. In this case, the incremental voltage—incrementalcurrent Equation 12 may be deduced:Δv=|Z _(L) +Z _(R) |·Δi _(Z)  Eq. 12

FIG. 5A illustrates a plot over time of the incremental voltage andincremental replica current for a forward fault consistent withembodiments of the present disclosure. Similarly, FIG. 5B illustrates aplot over time of the incremental replica voltage and incrementalreplica current for a reverse fault consistent with embodiments of thepresent disclosure. FIG. 5A and FIG. 5B illustrate that the incrementalreplica currents have similar waveforms and their relative polaritiesmay be used to indicate fault direction. Specifically, where thepolarity of Δv is opposite of the polarity of Δi_(Z), as in FIG. 5A, thefault may be a forward fault; and where the polarity of Δv is the sameas the polarity of Δi_(Z), as in FIG. 5A, the fault may be a reversefault. Further, the amplitude relationship between incremental voltageand incremental replica current depends on the system parameters and thefault direction.

Various embodiments consistent with the present disclosure may utilizeadaptive operating thresholds in connection with the directionalprotection principle. As described in connection with FIG. 5A and FIG.5B, the incremental voltage Δv and the incremental replica currentΔi_(Z) have opposite polarities for forward faults and of the samepolarity for reverse faults. Moreover, the peak amplitude of theincremental voltage equals |Z_(S)| times the peak amplitude of theincremental replica current for forward faults according to Equation 10,and it equals |Z_(L)+Z_(R)| times the peak amplitude of the incrementalreplica current for reverse faults, according to Equation 12. Thisrelationship may be used in certain embodiments to establish operatingquantities for incremental voltage and incremental replica current. Aninstantaneous operating quantity s_(OP) may be calculated from Equation13:s _(OP) =Δv·Δi _(Z)  Eq. 13

Substituting Equation 10 into Equation 13 yields an instantaneousoperating quantity s_(OP) for forward faults in Equation 14:s _(OP) =−|Z _(S)|·(Δi _(Z))²  Eq. 14

Similarly, substituting Equation 12 into Equation 13 yields aninstantaneous operating quantity s_(OP) for reverse faults in Equation15:s _(OP) =|Z _(L) +Z _(R)|·(Δi _(Z))²  Eq. 15

The instantaneous operating signal s_(OP) is negative for forward faultsand positive for reverse faults from the very first fault sample if theinput voltages and currents are sufficiently filtered. In oneembodiment, therefore, the directional element may determine thedirection to the fault (forward or reverse) by comparing the value ofs_(OP) against zero.

In another embodiment, the value of s_(OP) may be compared with twoadaptive thresholds, the forward adaptive threshold (s_(FWD)) and thereverse adaptive threshold (s_(REV)), defined using Equations 16 and 17to determine the direction to the fault:s _(FWD) =−Z _(FWD)·(Δi _(Z))²  Eq. 16s _(REV) =+Z _(REV)·(Δi _(Z))²  Eq. 17where values of Z_(FWD) and Z_(REV) are predetermined settings. Thevalues of Z_(FWD) and Z_(REV) may be similarly calculated using thesource and remote impedances using equations 18 and 19:Z _(FWD)=0.5·|Z _(S)|  Eq. 18Z _(REV)=0.5·|Z _(L)|  Eq. 19

Using equations 13-19, a forward fault may be indicated when Equation 20is satisfied:s _(OP) <s _(FWD)  Eq. 20

Furthermore, a reverse fault may be indicated when Equation 21 issatisfied:s _(OP) >s _(REV)  Eq. 21

Equations 20 and 21 are typically satisfied by samples during the faulttransient period. In certain embodiments, the operating quantity s_(OP)may be averaged and the adaptive thresholds s_(FWD) and s_(REV) may alsobe averaged over a time. Such embodiments may reduce noise in thesignal. The time window over which the signals are averaged may berelatively brief to facilitate rapid detection of faults. In variousembodiments, an IED consistent with embodiments of the presentdisclosure may use equations 16-21 to detect the direction of a fault inan electric power system. IEDs, such as those illustrated in FIG. 1, areoften configured to share information useful in protecting the electricpower delivery system.

FIG. 6 illustrates a plot over time of a direction element operation fora forward fault consistent with certain embodiments of the presentdisclosure. As expected for a forward fault, the operating signal(s_(OP)) is negative. The forward adaptive threshold (s_(FWD)), which iscalculated using Eq. 16, is negative and is equal to about half theoperating signal. The difference between the forward adaptive threshold(s_(FWD)) and the operating signal (s_(OP)) may be referred to as adependability margin.

The operating signal (s_(OP)), the reverse adaptive threshold (s_(REV)),and the forward adaptive threshold (s_(FWD)) are not averaged in thisexample. In embodiments in which averaging is used, the averaging wouldreduce the rate of change of these signals, but would not affect therelationship between the operating signal and the thresholds. Based onthe conditions illustrated in FIG. 6, systems and methods consistentwith the present disclosure would identify a forward fault.

In various embodiments, systems and methods consistent with the presentdisclosure may rely on one-end voltage and current measurements, and assuch, may not need to receive communications or quantities from anotherIED to determine fault direction using time-domain quantities. In oneembodiment, an IED may include a directional module using thedirectional element configured to determine the direction of a faultbased on Eq. 16 through Eq. 21 above, and as described in connectionwith FIG. 6. Such a directional module may be configured to operate onlyin an area of protection that falls short of a remote terminal. Such aconfiguration may be referred to herein as “underreach.”

In various embodiments, various techniques may be utilized to avoid“overreach” of the designated area of protection. Such a directionalmodule may further include transient overreach control. One suchtechnique may be illustrated with reference to the system illustrated inFIG. 4B. In FIG. 4B, the voltage at the fault point may be representedby Equation 22:Δv−m·|Z _(L) |·Δi _(Z) =ΔV _(F)  Eq. 22In Eq. 22, m represents a distance to the fault normalized for the linelength. The distance module may be configured to “reach” up to a certainpoint (m=m₀) on the protected line, where m₀ is short of the remote bus.Accordingly, the distance module may not “reach” faults beyond m₀. Fromthe Thévenin's theorem, the highest fault voltage ΔV_(F) is the systemvoltage (V_(SYS)) plus some margin for possible pre-fault overvoltage.In some embodiments, the system voltage is the nominal system voltage.Thus, selective tripping may be calculated using Equation 23:|ΔV _(F) |>k ₀ ·V _(SYS)  Eq. 23where k₀ is greater than unity. In one embodiment, k₀ may be 1.1.

Using equations 22 and 23, an operating equation for an underreachconfiguration may be calculated using Equations 24A and 24B:S _(OP) _(_) _(U) =|Δv−m ₀ ·|Z _(L)|·Δ_(iz)|  Eq. 24AS _(OP) _(_) _(U) >k ₀ ·V _(SYS)  Eq. 24BUsing Equation 24A, an underreaching element may issue a trip command ifthe calculated voltage of the underreach operating signal, s_(OP) _(_)_(U), at the intended reach point m₀ is greater than k₀·V_(SYS), asstated in Eq. 24B. In one embodiment, the value of system voltageV_(SYS) may be a setting specified by a user.

FIG. 7A illustrates a plot over time of an incremental voltage, for anin-zone fault occurring near the voltage peak and located atapproximately 60% of the set reach point of a directional elementconsistent with one embodiment of the present disclosure. FIG. 7B andFIG. 7C illustrate plots over time of an incremental replica current andan operating signal, respectively, under the same conditions illustratedin FIG. 7A. The quantity k₀·V_(SYS) is illustrated in FIG. 7C, and thepoint at which s_(OP) _(_) _(U) crosses the quantity k₀·V_(SYS), anunderreach element may issue a trip command.

The voltage at the fault point collapses very steeply during linefaults. As a result, the left-hand side of Eq. 24, which may correspondto a measurement of the value of |Δv_(F)|, reflects the step change ofthe fault point voltage. Accordingly, Eq. 24 becomes satisfied quicklyfor in-zone faults occurring near the voltage peak, as illustrated inFIG. 7C. In the illustrated embodiment, the initial rise of the value ofs_(OP) _(_) _(U) is slowed down by a digital low-pass filter. A low-passfilter may facilitate the use of the RL line and system modelillustrated in FIG. 4. By changing the cutoff frequency of the filter,the balance between speed and security may be controlled.

In certain electric power delivery systems, the voltage of the systemmay differ from the nominal voltage. For example, a utility may operatean electric power system at a voltage of 110% of the nominal voltage.Such an increase may be beneficial in that more power may be transferredwith less loss on a percentage basis. However, such operation mayincrease the risk of faults or damage to equipment. Utilities mayconversely operate an electric power system below the nominal voltage.Although this may increase the percentage of power loss in transmission,it may also decrease the risk of fault or equipment damage.

In one embodiment, a setting is used to compensate for operation of theelectric power system at a voltage that is not the nominal voltage, thesystem voltage V_(SYS) used in Equation 24 may be calculated usingmeasured system voltages. In such embodiments, the system voltage may bea root-mean square value of measured voltage. The voltage may be anaverage of the absolute peak values of the voltage. The system voltagemay be continuously measured and updated during steady state or duringnon-fault events. The system voltage may be measured and updated on aperiodic basis such as once a cycle or once every few cycles, or thelike.

Equation 24 may be re-arranged to produce Equation 25:

$\begin{matrix}{{{{\frac{\Delta\; v}{m_{0}} - {{{Z_{L}} \cdot \Delta}\; i_{Z}}}} > \frac{k_{0} \cdot V_{SYS}}{m_{0}}}{or}{m_{0} > \frac{k_{0} \cdot V_{SYS}}{{\frac{\Delta\; v}{m_{0}} - {{{Z_{L}} \cdot \Delta}\; i_{Z}}}}}} & {{Eq}.\mspace{14mu} 25}\end{matrix}$

In certain embodiments, equations representing the underreaching element(e.g., Eqs. 24-25) may be used in a directional module to determine adirection to a fault and to determine whether the fault is within a zoneof protection of the directional module using time-domain quantitieswithout measurements from another IED. The directional module may alsouse Equations 16-21 to determine the direction to the fault. In oneembodiment, the directional module determines a direction to the faultusing Equations 16-21. If the fault is in the forward direction, thedirectional module may determine whether the fault is within apredetermined zone of protection using Equations 24-25. If, the fault iscalculated to be in the forward direction and within the predeterminedzone of protection, then the directional element may issue a protectioncommand. For example, the IED with the directional element may thenissue an open command to a circuit breaker or take a similar action toprotect the line from the fault.

FIG. 8 illustrates an operating characteristic of an underreachingelement with directional supervision consistent with certain embodimentsof the present disclosure. In the illustrated embodiment, the x-axisrepresents Δv and the y-axis represents |Z_(L)|·Δi_(Z). Equation 25 isplotted in FIG. 8, together with an alternate representation of Eq. 10.Eq. 10 may also be expressed in terms of the product of the lineimpedance and the replica current, as shown in Eq. 26.

$\begin{matrix}{{\Delta\; v} = {{{{- {Z_{s}}} \cdot \Delta}\; i_{Z}} = {{{{- \frac{Z_{S}}{Z_{L}}} \cdot {Z_{L}} \cdot \Delta}\; i_{Z}} = {{{- {SIR}} \cdot {Z_{L}} \cdot \Delta}\; i_{Z}}}}} & {{Eq}.\mspace{14mu} 26}\end{matrix}$In Eq. 26, SIR is the source-to-line impedance ratio (i.e.,|Z_(s)|/|Z_(L)|), which is a quantity that may range between a highvalue SIR_(MAX) and a low value SIR_(MIN).

FIG. 9 illustrates an operating characteristic of an overreachingnondirectional element consistent with certain embodiments of thepresent disclosure. In various embodiments, an overreaching element maybe configured to selectively enable or disable a directional element. Anoperating characteristic of one embodiment of an overreaching elementmay be determined based on Eq. 24. The overreaching nondirectionalelement may reach beyond a remote line terminal up to the m1 point (m1>1pu) and to respond to faults that generate a relatively small change inthe voltage. As such, the operating characteristic may be specified byEq. 27.|Δv−m ₁ ·|Z _(L) |·Δi _(Z) |>k ₁ ·V _(SYS)  Eq. 27

In yet another embodiment, a directional module may determine whether afault is in the forward or reverse direction using the time-domainquantities of incremental voltage Δv and incremental current Δi. For ashort initial period of time after a fault (or other step-change), thecurrent change and voltage change may be related to the fault direction.The current change and voltage change are of opposite polarities forforward faults, and the same polarity for reverse faults. Thus, in oneembodiment, the directional module may compare the polarities of theincremental voltage and the incremental current. When the incrementalvoltage and incremental current are of the same polarity, the distancemodule may determine that the fault is in the forward direction, andenable a fault detection and location calculation module. When theincremental voltage and incremental current are of the same polarity,however, the distance module may determine that the fault is in thereverse direction and block the fault detection and location calculationmodule.

After a short initial time from the fault, the electric power systemchanges from exhibiting purely resistive behavior to theresistive-inductive behavior. At such time, the incremental replicacurrent Δi_(Z) begins to describe the system better than does theincremental current Δi. Accordingly, in some embodiments, a directionalelement may utilize detection techniques appropriate to determine afault direction based on the resistive behavior present for a shortwindow of time following a fault, or based on the resistive-inductivebehavior that persists for longer periods of time. In one embodiment,the directional module may include a short time window (e.g., a windowof 1 ms or 2 ms) during which the incremental current (instead of theincremental replica current) may be used to determine direction to thefault by comparison of the incremental current polarity against thepolarity of the incremental voltage. After the window of time, thedirectional element may switch to using equations 16-21 to determine thedirection to the fault based on the incremental quantities that betterreflect the resistive-inductive behavior of the network.

In electric power systems that include multiple phases (e.g.,three-phase alternating current electric power delivery systems), afault may involve more than one phase and/or may affect more than onephase. That is, in multi-phase systems, several fault types arepossible, including single-phase-to-ground faults, phase-to-phasefaults, phase-to-phase-to-ground faults, three-phase faults, and thelike. In several embodiments herein, the fault detection and locationcalculation may be independent from the fault type in the actualmultiple-phase power system.

For a three-phase (A, B, and C) power system, in the frequency domain,an A-phase to ground (AG) fault voltage drop in the A phase may berepresented by Equation 28:V _(A) =Z ₁ I ₁ +Z ₂ I ₂ +Z ₀ I ₀  Eq. 28where:

V_(A) is the voltage drop in the A phase;

Z₁ is the positive-sequence impedance;

I₁ is the positive-sequence current;

Z₂ is the negative-sequence impedance;

I₂ is the negative-sequence current;

Z₀ is the zero-sequence impedance; and

I₀ is the zero-sequence current.

Equation 28 may be re-arranged to obtain a relationship between thefaulted phase voltage, the positive-sequence impedance and a new currentI_(AG) referred to as the “loop current” in Equation 29:V _(A) =Z ₁ I _(AG)  Eq. 29

The loop current that makes Equation 28 conform with the format ofEquation 29 is shown in Equation 30:

$\begin{matrix}{I_{AG} = {{{I_{A} \cdot 1}{\angle\Theta}_{1}} - {{I_{0} \cdot 1}{\angle\Theta}_{1}} + {\frac{Z_{0}}{Z_{1}}{I_{0} \cdot 1}{\angle\Theta}_{0}}}} & {{Eq}.\mspace{14mu} 30}\end{matrix}$where:

I_(A) is the A-phase current; and

Θ₀ and Θ₁ are the angles of the zero- and positive-sequence lineimpedances (Z₀ and Z₁), respectively.

Returning to the replica current of Equation 8, at nominal systemfrequency, the replica current Δi_(Z) is a voltage drop across an RLcircuit with a gain of one. Thus, Equation 30 may be restated with thatsubstitution as Equation 31:

$\begin{matrix}{{f_{IZ}( {{\Delta\; i},R,L} )} = {{{{D_{0}( {R,L} )} \cdot \Delta}\; i} + {{{D_{1}( {R,L} )} \cdot \frac{\mathbb{d}\;}{\mathbb{d}t}}\Delta\; i}}} & {{Eq}.\mspace{14mu} 31}\end{matrix}$where D₀ and D₁ are given by Equation 9. Equation 28 may thus be used tocalculate the loop current in the time domain using Equation 30:

$\begin{matrix}{i_{AG} = {{f_{IZ}( {i_{A},R_{1},L_{1}} )} - {f_{IZ}( {i_{0},R_{1},L_{1}} )} + {\frac{Z_{0}}{Z_{1}}{f_{IZ}( {i_{0},R_{0},L_{0}} )}}}} & {{Eq}.\mspace{14mu} 32}\end{matrix}$where R₁ and L₁ and R₀ and L₀ are the resistance and inductance of thepositive- and zero-sequence line impedances.

Equations 33-37 calculate several quantities that may be used in thecalculation of loop voltages and currents:

$\begin{matrix}{{\Delta\; i_{0}} = {\frac{1}{3}( {{\Delta\; i_{A}} + {\Delta\; i_{B}} + {\Delta\; i_{C}}} )}} & {{Eq}.\mspace{14mu} 33} \\{{\Delta\; i_{0Z}} = {{f_{IZ}( {i_{0},R_{1},L_{1}} )} - {\frac{Z_{0}}{Z_{1}}{f_{IZ}( {i_{0},R_{0},L_{0}} )}}}} & {{Eq}.\mspace{14mu} 34} \\{i_{AZ} = {f_{IZ}( {i_{A},R_{1},L_{1}} )}} & {{Eq}.\mspace{14mu} 35} \\{i_{BZ} = {f_{IZ}( {i_{B},R_{1},L_{1}} )}} & {{Eq}.\mspace{14mu} 36} \\{i_{CZ} = {f_{IZ}( {i_{C},R_{1},L_{1}} )}} & {{Eq}.\mspace{14mu} 37}\end{matrix}$

Quantities in Equations 33-37 may be used to calculate loop voltages andcurrents according to Table 1:

Loop Voltage Current AG Δν_(A) Δi_(AZ) − Δi_(0Z) BG Δν_(B) Δi_(BZ) −Δi_(0Z) CG Δν_(C) Δi_(CZ) − Δi_(0Z) AB Δν_(A) − Δν_(B) Δi_(AZ) − Δi_(BZ)BC Δν_(B) − Δν_(C) Δi_(BZ) − Δi_(CZ) CA Δν_(C) − Δν_(A) Δi_(CZ) −Δi_(AZ)

Calculation of a fault and fault location may then be performed forspecific fault type by using the correct voltage and current quantitiesas calculated using Table 1. For example, a directional module maycalculate an underreach condition using Equation 24, where theincremental voltage in the time domain Δv and the incremental replicacurrent Δi_(Z) used for the calculation is the appropriate incrementalvoltage for the fault type from Table 1, and the incremental replicaΔi_(Z) current used is the appropriate incremental replica current fromTable 1. That is, if the fault type is an A-phase-to-ground fault, theincremental voltage used is Δv_(A), and the incremental replica currentused is Δi_(AZ)−Δi_(0Z). If the fault type is an A-phase-to-B-phasefault, the incremental voltage used is Δv_(A)−Δv_(B), and theincremental replica current used is Δi_(AZ)−Δi_(BZ).

Selection of the appropriate faulted phases for use of the appropriatefault loop quantities, above, may require identification of the faultedphases. According to one embodiment, to quickly determine which phase orphases are the faulted phase or phases, the relative incremental currentquantities of each phase may be made. The phase(s) experiencing thegreatest relative incremental current quantities may be determined asthe faulted phase(s).

In another embodiment, the operating signals, such as those calculatedusing Equation 13 calculated for each of the loops allows for selectingthe faulted phases. For example, the operating quantities for each loopare calculated as shown in Table 2:

Loop S_(OP) = AG Δν_(A) * (Δi_(AZ) − Δi_(0Z)) BG Δν_(B) * (Δi_(BZ) −Δi_(0Z)) CG Δν_(C) * (Δi_(CZ) − Δi_(0Z)) AB (Δν_(A) − Δν_(B)) * (Δi_(AZ)− Δi_(BZ)) BC (Δν_(B) − Δν_(C)) * (Δi_(BZ) − Δi_(CZ)) CA (Δν_(C) −Δν_(A)) * (Δi_(CZ) − Δi_(AZ))

Once the operating quantities are calculated for each loop, theoperating quantities may be compared. The loop with the highestoperating quantity is the faulted loop. Once the faulted loop isdetermined, the incremental voltage and incremental current quantitiesassociated with that loop (from Table 1) may be used to determineunderreach and directional quantities. The above techniques may be usedby an IED such as IEDs 102, 104 of FIG. 1 to determine the direction toa fault and whether the fault is in a zone of protection.

FIG. 10 illustrates a flowchart of one embodiment of a method 1000 fordetermining a direction to a fault using time-domain quantitiesconsistent with embodiments of the present disclosure. The method 1000starts by receiving electric power system signals at 1002 andcalculating therefrom incremental quantities (e.g., Δi_(Z) and Δv) at1004. Based on the incremental qualities, method 1000 may then determinethe fault type at 1006. In a three phase system, the fault type may bedetermined using equations 33-37. The method may then calculate anoperating quantity for the fault type at 1008. The operating quantitymay be used to determine a fault direction. The operating quantity maybe calculated using, for example, equations 13-15, and may use theincremental quantities of Table 1 depending on the fault type. That is,the operating quantity may be calculated using the equations of Table 1.The direction (forward or reverse) to the fault may then be determinedat 1010 using the operating quantities shown in Table 1. In someembodiments, the method 1000 may use incremental quantities to calculatethe operating quantity directly using Equation 13, without using theincremental quantities of Table 1.

If the fault is a forward fault, at 1012, method may end, and otherwise,method 1000 may determine at 1020 whether the fault is within the zoneof protection. Any of the techniques disclosed herein for assessing azone of protection may be used in various embodiments. If the fault isdetermined to be within the zone of protection, at 1020, the fault maybe cleared at 1022.

FIG. 11 illustrates a flow diagram of one embodiment of a method 1100for determining whether a fault is within a zone of protectionconsistent with the present disclosure. The method 1100 starts bycalculating the highest possible fault voltage at 1102 using, forexample, Equation 23. In some embodiments, the highest possible faultvoltage may be calculated once and stored for later use. Alternatively,this may be calculated on a periodic basis using quantities measuredfrom the electric power system as described herein. The method proceedswith receipt of electric power system signals at 1104. Using thesignals, the method then calculates incremental quantities (e.g., Δi_(Z)and Δv) at 1106. The method may then determine the fault type using theincremental quantities at 1108. The fault type may be determined usingthe operating quantities listed in Table 2. The fault voltage at theintended reach point may then be calculated at 1110 using theappropriate incremental quantities according to the determined faulttype as shown in Table 1. The fault voltage at the intended reach pointmay then be compared with a scaled highest possible voltage at 1112. Ifthe fault voltage at the intended reach point is greater than the scaledhighest possible voltage, then underreach is determined and a fault maybe declared.

FIG. 12 illustrates one embodiment of a method for determining adirection to a fault and for determining whether the fault is within azone of protection consistent with certain embodiments of the presentdisclosure. Method 1200 starts with receiving electric power systemsignals at 1202. As described above, for a predetermined period thedirection may be determined using incremental current during a timewhere the power system exhibits resistive behavior. Thus, the method maydetermine whether the power system exhibits resistive behavior at 1204.This determination may be made by calculating a time from the instanceof the fault and during a predetermined time period the system may beassumed to be exhibiting resistive behavior. If the system is withinthat time period, then the method may calculate incremental current andincremental voltage as quantities Δi and Δv at 1206. If, however, anevent occurs outside of the resistive behavior period, method 1200 maycalculate incremental quantities as Δi_(Z) and Δv 1208. The method maythen calculate an operating quantity 1210 using the incrementalquantities determined at either 1206 or 1208. Method 1200 may determinethe fault type at 1211. Fault type determination may be performed bycomparing relative incremental current quantities from each phase asdescribed above.

With the operating quantity from 1210, method 1200 may then determinethe direction to the fault 1212 using, for example, equations 16-21. If,at 1214, the fault is determined to be in the reverse direction (i.e.,the fault is not a forward fault), If, however, the method determinesthat the fault is in the forward direction at 1214, method 1200 proceedsto forward the fault direction information to a remote relay at 1216.The remote relay may utilize the information in connection withdetermining whether the fault is within a zone of protection.

At 1218, method 1200 may calculate fault voltage at the intended reachpoint. With the calculated fault voltage, method 1200 may determineunderreach by comparing the fault voltage at the intended reach pointwith a scaled highest possible voltage at 1220. If it is determined thatthe fault is within the zone of protection at 1221 (the voltage at theintended reach point is greater than the scaled highest possiblevoltage), then the method may clear the fault at 1222. In variousembodiments, the fault may be cleared by, for example, opening abreaker. After the fault is cleared, the method may end.

Fault location according to several embodiments herein may be performedusing or in connection with a traveling wave differential module. Thetraveling wave differential module described herein overcomes therequirements for high-fidelity voltage signals of previous travelingwave fault protection techniques. In some embodiments consistent withthe present disclosure, a traveling wave differential module may operateusing a current-only traveling wave differential scheme by comparingvalues of the current waves, without isolating the incident andreflected waves as in previous traveling wave location techniques. Inother embodiments, voltage signals associated with traveling waves mayalso be analyzed to identify a fault location.

Current waves as measured by the traveling wave differential module maybe sums of the incident and reflected waves. For external faults theamplitudes of the current waves would not match between the lineterminals because of the termination effects. Isolating incident andreflected waves makes the measurement independent of the terminationimpedances, but it requires high-fidelity voltage information.

However, the total current waves retain the polarity of the incidentwaves. Therefore, the amplitudes of the total current waves at both lineterminals may be compared, taking into account the line propagationdelay. For a healthy line the polarities will be inverted. Thus, thetraveling wave differential module may determine if the fault isinternal to the line using currents by making assumptions of faultdirection, and checking amplitudes of the traveling waves. According toone embodiment, the traveling wave differential module may assume aninternal fault, and calculate the fault current by aligning and addingthe amplitudes of the first current traveling waves that arrived at thelocal and remote terminals respectively using, for example, Equation 38:i _(OP(t)) =|i _(S(t)) +i _(R(t−P))|  Eq. 38where:

i_(OP(t)) is the operating current at time t;

i_(S(t)) is the current at local Terminal S at the time t when thetraveling wave arrives at the local Terminal S; and

i_(R(t−P)) is the current at remote terminal R at the time (t−P) whenthe traveling wave arrives at the remote terminal R, where P is thedelay between the traveling wave arrival at the remote terminal and thelocal terminal. P T, where T is the line propagation delay between thelocal and remote terminals.

Equation 38 assumes that the traveling wave arrived at the remote (R)terminal before it arrived at the local (S) terminal. If, however, thetraveling wave first arrives at the local (S) terminal, Equation 39 maybe used:i _(OP(t)) =|i _(S(t−P)) +i _(R(t))|  Eq. 39where:

i_(S(t−P)) is the current at local Terminal S at the time (t−P) when thetraveling wave arrives at the local Terminal S, where P is the delaybetween the traveling wave arrival at the local terminal and the remoteterminal, and P≦T, where T is the line propagation delay between thelocal and remote terminals; and

i_(R(t)) is the current at remote Terminal R at the time (t) when thetraveling wave arrives at the remote Terminal R.

Either Equation 38 or 39 may be used depending on which terminal firstreceived the traveling wave. Equations 38 and 39 may be executed onlyonce, giving a single value of i_(OP) that reflects (for internalfaults) the total traveling wave fault current. For internal faults, thetraveling wave polarities at the remote and local terminals are thesame, yielding a high value when summed as in equations 38 and 39.

If the sum of the two current traveling waves is significant, then acheck is made to determine that the fault is not external. In the check,the traveling wave differential module may assume an external fault, andcalculates the passing through current traveling wave that would result.For external faults, the traveling wave that entered at one lineterminal leaves at the other terminal after the line propagation timedelay T. The check may be made by calculating current restraintquantities as shown in Equations 40 and 41:i _(RT1(t)) =|i _(S(t)) −i _(R(T−T))|  Eq. 40i _(RT2(t)) =|i _(R(t)) −i _(S(t−T))|  Eq. 41Equation 40 may be executed once at the point in time t after the firsttraveling wave arrived at the remote Terminal R. Similarly, Equation 41may be executed once at the point in time t after the first travelingwave arrived at the local Terminal S. The restraint quantities may becombined using, for example, Equations 42 or 43:

$\begin{matrix}{i_{RT} = {\max( {i_{{RT}\; 1},i_{{RT}\; 2}} )}} & {{Eq}.\mspace{14mu} 42} \\{i_{RT} = {\frac{1}{2}( {i_{{RT}\; 1} + i_{{RT}\; 2}} )}} & {{Eq}.\mspace{14mu} 43}\end{matrix}$Finally, the traveling wave differential module may declare a fault bycomparing the operating and restraint quantities. Equation 44 may beused for such comparison:i _(OP) >k·i _(RT)  Eq. 44where:

k is a restraining factor.

In various embodiments, k comprises a user-specified factor.

In some embodiments, a traveling wave differential module consistentwith the present disclosure may operate using representations ofelectrical conditions from terminals on opposite sides of a transmissionline, where the current quantities include time stamps. The travelingwave differential module may time align the time stamped currentquantities from each terminal.

FIG. 13 illustrates a block diagram of a system configured to use of acorrelation technique for determining a fault location consistent withcertain embodiments of the present disclosure. In the embodimentillustrated in FIG. 13, a traveling wave differential module may usemultiplication instead of signal addition. This approach is known in theindustry as a correlation technique. Operating and restraint quantity iscalculated as:i _(OP(t)) =i _(S(t−P)) *i _(R(t))  Eq. 45where:

i_(S(t−P)) is the current at local terminal S at the time (t−P) when thetraveling wave arrives at the local terminal S, where P is the delaybetween the traveling wave arrival at the local terminal and the remoteterminal, and P≦T, where T is the line propagation delay between thelocal and remote terminals; and

i_(R(t)) is the current at remote Terminal R at the time (t) when thetraveling wave arrives at the remote Terminal R.

And the restraint current:i _(RT)=|max(i _(S(t−P)))|+|max(i _(R(t)))|  Eq. 46Finally, the traveling wave differential module may declare a fault bycomparing the operating and restraint quantities. Equation 47 may beused for such comparison:i _(OP) >k·i _(RT)  Eq. 47where:

k is a restraining factor.

Correlation calculations can be performed once upon detection of thetraveling wave signal or continuously for every input signal sample.Correlator output may further be filtered by averaging a number ofoutput results (samples) with filter length adjusted to encompass singlewave peaks as shown in FIGS. 14B, 14C, and 14D. Operation of thecorrelator described by Eq. 45 may further be modified to search for thetime delay P; using P as the unknown variable and performingcalculations for all time delays in the range P≦T (all delays lower thanthe line propagation delay T).

Correlation technique can also be applied to continuous monitoring ofthe transmission line health. In this approach transmission line issubdivided into a number of segments with a separate correlator assignedto each segment as shown in FIG. 13. One objective of such an approachis to detect any energy that may be originating from a particular linesegment. This energy may include excessive corona discharge, partialinsulation breakdown and localized insulator arcing that may be presentbefore the fault and lightning strikes in the immediate vicinity of theline. Line activity is monitored continuously with high frequency energyoriginating in each segment accumulated over a selectable time period ascommanded by the Time Trigger signal (for example 1 second to 24 hours).Accumulated data is subsequently stored for further statistical analysisand alarming purposes.

As illustrated in FIG. 13, individual correlators are fed by twosignals. The remotely measured signal (current or voltage for a givenphase) obtained through communications and a delayed version of thelocally measured signal on the same phase. Each correlator may receive adifferent delay such that the signals generated on a given segment onthe transmission line are lined up to the selected correlator inputsregardless of the fact they are measured on different ends of thetransmission line.

Any number of correlators (observed line segments) may be linked to thesampling frequency used to perform measurements at the two ends of theline. For example, with a sampling frequency set to 1 MHz, and the knownpropagating speed of the traveling wave signals (close to the speed oflight c=299.8e6 m/s), the traveling wave will travel 299.8 m (shown as300 m herein) between two consecutive samples. If the correlator delaysare set one sample (1 μs) apart, spatial resolution becomes equal to onehalf of the travel time 300 m/2=150 m. The number of correlatorsrequired to cover the entire line length can be calculated according to:

$\begin{matrix}{n = {2*L*\frac{fs}{c}}} & {{Eq}.\mspace{14mu} 48}\end{matrix}$where:

n is the number of correlators.

L is the line length

c is the traveling wave propagating speed which is close to the speed oflight

fs is the sampling frequency

FIG. 13 also shows a real time “Maximum Search” component preferablyrunning at the correlator rate (i.e. 1 MHz). This component is taskedwith finding the highest correlator output in real time, and reportingit as a possible fault location candidate. Since each correlator isassociated with a particular segment of the line; highest outputassociated with the traveling wave arrival directly identifies exactlocation of the power system fault that caused the traveling wave.

In various embodiments, the sampling frequency may be adjusted (higheror lower than 1 MHz), with the total number of correlators selected tomeet the desired spatial resolution. Individual correlators can beassigned to individual transmission line phases (A, B, C) multiplyingthe total number of correlators required to cover the line by 3.

FIG. 14A illustrates TWs launched by a fault 1402 and reflected atlocations U, S, and R consistent with certain embodiments of the presentdisclosure. The TW launched by the fault is reflected at the relaylocation S, is reflected again at the fault point 1402, and returns tolocation S. An underreaching TW distance element may be designed bymeasuring the time difference Δt between the arrival of the first TWfrom the fault and the arrival of the TW reflected at the fault point.The element calculates the fault distance using Δt and the wavepropagation velocity and issues a trip if the distance is shorter thanthe set reach.

In one embodiment, the distance to the fault may be calculated and anappropriate control action may be initiated using the following steps.First, upon arrival of the first TW to the line terminal, a faultdetection system determines the fault direction using a directionalelement, as disclosed herein. For faults in the forward direction, anestimate Δt may be determined between the two TWs, as shown in FIG. 14A.Further, a system may use cross-correlation to verify similarity of thewave reflected from the fault and the prior wave traveling toward thefault. Third, the system may calculate the distance to the fault usingEq. 49.

$\begin{matrix}{d = {\frac{\Delta\; t}{2} \cdot v}} & {{Eq}.\mspace{14mu} 49}\end{matrix}$Fourth, if d is less than a reach setting, a control action may beinitiated to clear the fault.

FIG. 14B illustrates a plot over time of traveling wave currentsreceived at Terminal S and Terminal R in FIG. 14A during an externalfault consistent with certain embodiments of the present disclosure. Ineach of FIGS. 14B, 14C and 14D, the Terminal S signal (solid line) andTerminal R (dashed line) signal illustrate the B-phase alpha current(B-phase current minus a zero-sequence current). The TWs illustrated inthese figures may be obtained using a differentiator-smoother filterhaving a window length of 20 microseconds. The current TW entered theprotected line at the local terminal at 30.20 milliseconds with a valueof around +462 A and left the line at the remote terminal at 31.23milliseconds with a value of around −464 A. The operating signalcalculated is approximately 2 A. The restraining signal is approximately926 A. The restraining signal (926 A) is much greater than the operatingsignal (2 A), therefore, the element would be restrained and the elementrestrains as expected.

FIG. 14C illustrates a plot over time of traveling wave currentsreceived at Terminal S and Terminal R in FIG. 14A during an externalfault consistent with certain embodiments of the present disclosure. Asa result, the first TWs arrived at Terminal S and Terminal R with thesame polarity and only around 0.2 milliseconds apart. The operatingsignal calculated for P=0.2 milliseconds equals around 403 A+219 A=622A. This case could be mistaken for an internal fault. Notice that theTerminal S TW that entered at around 30.50 milliseconds with anamplitude of 403 A left Terminal R at around 31.53 milliseconds with anamplitude of −411 A. Similarly, the TW that entered Terminal R at around30.65 milliseconds with an amplitude of 219 A left Terminal S at 31.68milliseconds with an amplitude of −208 A. Therefore, the restrainingsignals are 403−(−411)=814 A and 219−(−208)=427 A. The total restrainingsignal per Equation 40 is 814 A. Because the restraining signal (814 A)is greater than the operating signal (622 A), the element restrains asexpected, (using k=1).

FIG. 14D illustrates a plot of traveling wave currents received atTerminal S and Terminal R in FIG. 14A during an internal faultconsistent with certain embodiments of the present disclosure. Theoperating signal for this case is around 960+785=1745 A. The restrainingsignals are around 960 A and 785 A, respectively, because the initialwaves do not leave the line after the line propagation time. The totalrestraining signal is therefore around 960 A. As a result, the operatingsignal (1,745 A) is much greater than the restraining signal (960 A),and the element operates dependably.

The signals illustrated in FIGS. 14B-14C may be acquired in someembodiments using a configuration similar to what is illustrated inFIG. 1. Specifically, IEDs 102 and 104 may be in electricalcommunication with the local 112 and remote 114 terminals of theelectric power system, and in communication with each other usingcommunication channel 108. In one embodiment including a traveling wavedifferential module, communication channel 108 may utilize a high-speedcommunication channel allowing for exchange of electrical currentmeasurements from each of the IEDs 102, 104 at a rate between about500,000 and about 5,000,000. In one specific embodiment, themeasurements may be exchanged at a rate of around 1 millionsamples-per-second (1 Msps). Such high-sampling rates may enable acontrol system to detect and respond to faults more quickly than lowerfrequency sampling rates.

FIG. 15 illustrates a flow diagram of a method for determining internalfault conditions using a traveling wave differential module consistentwith certain embodiments of the present disclosure. In some embodiments,method 1500 may be implemented using only measurements of electricalcurrents in an electric power distribution system. Method 1500 startswith receiving local and remote current quantities at 1502. As discussedhereinabove, IEDs may be configured to receive electric power systemmeasurements at two terminals of an electric power delivery system, andto share current quantities at a rate sufficient for detection oftraveling waves. In order to accommodate the transitory nature oftraveling waves, systems and methods consistent with the presentdisclosure may be configured to process and/or share information at arate of approximately 1 million measurements per second. In variousembodiments, the method 1500 may either time align the local and remotecurrent quantities at 1504, or correlate local and remote currentquantities at 1506. In some embodiments, the method may both time alignand correlate local and remote current quantities at 1504 and at 1506.Using the time aligned or correlated current quantities, the method maythen calculate a traveling wave operating quantity 1508 using, forexample, equations 38-39. Method 1500 may then calculate a travelingwave restraint quantity 1510 using, for example equations 40-43. Themethod may then compare the operating and restraint quantities at 1512.With the comparison, the method may then determine whether the fault isan internal fault at 1514. If the fault is an internal fault, the faultmay be cleared at 1516 through a protection (e.g., opening a circuitbreaker). If the fault is not an internal fault, method 1500 may end.

FIG. 16 illustrates a functional block diagram of a system 1600 fordetecting and locating faults using time-domain quantities consistentwith embodiments of the present disclosure. In certain embodiments, thesystem 1600 may comprise an IED system configured to, among otherthings, obtain and calculate time-domain quantities, detect and locatefaults using a time-domain distance module, detect and locate faultsusing a time-domain directional module, and detect and locate faultsusing traveling waves. System 1600 may be implemented using hardware,software, firmware, and/or any combination thereof. In some embodiments,system 1600 may be embodied as an IED, while in other embodiments,certain components or functions described herein may be associated withother devices or performed by other devices. The specificallyillustrated configuration is merely representative of one embodimentconsistent with the present disclosure.

System 1600 includes a communications interface 1616 configured tocommunicate with devices and/or IEDs. In certain embodiments, thecommunications interface 1616 may facilitate direct communication withother IEDs or communicate with systems over a communications network.Communications interface 1616 may facilitate communications through anetwork. System 1600 may further include a time input 1612, which may beused to receive a time signal (e.g., a common time reference) allowingsystem 1600 to apply a time-stamp to the acquired samples. In certainembodiments, a common time reference may be received via communicationsinterface 1616, and accordingly, a separate time input may not berequired for time-stamping and/or synchronization operations. One suchembodiment may employ the IEEE 1588 protocol. A monitored equipmentinterface 1608 may be configured to receive status information from, andissue control instructions to, a piece of monitored equipment (such as acircuit breaker, conductor, transformer, or the like).

Processor 1624 may be configured to process communications received viacommunications interface 1616, time input 1612, and/or monitoredequipment interface 1608. Processor 1624 may operate using any number ofprocessing rates and architectures. Processor 1624 may be configured toperform various algorithms and calculations described herein. Processor1624 may be embodied as a general purpose integrated circuit, anapplication specific integrated circuit, a field-programmable gatearray, and/or any other suitable programmable logic device.

In certain embodiments, system 1600 may include a sensor component 1610.In the illustrated embodiment, sensor component 1610 is configured togather data directly from conventional electric power system equipmentsuch as a conductor (not shown) using conventional PTs and/or CTs. Thesensor component 1610 may use, for example, transformers 1602 and 1614and A/D converters 1618 that may sample and/or digitize filteredwaveforms to form corresponding digitized current and voltage signalsprovided to data bus 1622. Current (I) and voltage (V) inputs may besecondary inputs from conventional instrument transformers such as, CTsand VTs. A/D converters 1618 may include a single A/D converter orseparate A/D converters for each incoming signal. A current signal mayinclude separate current signals from each phase of a three-phaseelectric power system. A/D converters 1618 may be connected to processor1624 by way of data bus 1622, through which digitized representations ofcurrent and voltage signals may be transmitted to processor 1624. Invarious embodiments, the digitized current and voltage signals may beused to calculate time-domain quantities for the detection and thelocation of a fault on an electric power system as described herein.

A computer-readable storage medium 1626 may be the repository of adatabase 1628 containing electric power line properties for eachtransmission line and/or each section of each transmission line, such asimpedances, resistances, propagation times, reactances, lengths, and/orthe like. Another computer-readable storage medium 1630 may be therepository of various software modules configured to perform any of themethods described herein. A data bus 1642 may link monitored equipmentinterface 1608, time input 1612, communications interface 1616, andcomputer-readable storage mediums 1626 and 1630 to processor 1624.

Computer-readable storage mediums 1626 and 1630 may be separate mediums,as illustrated in FIG. 16, or may be the same medium (i.e. the samedisk, the same non-volatile memory device, or the like). Further, thedatabase 1628 may be stored in a computer-readable storage medium thatis not part of the system 1600, but that is accessible to system 1600using, for example, communications interface 1616.

Communications module 1632 may be configured to allow system 1600 tocommunicate with any of a variety of external devices via communicationsinterface 1616. Communications module 1632 may be configured forcommunication using a variety of data communication protocols (e.g., UDPover Ethernet, IEC 61850, etc.).

Data acquisition module 1640 may collect data samples such as thecurrent and voltage quantities and the incremental quantities. The datasamples may be associated with a timestamp and made available forretrieval and/or transmission to a remote IED via communicationsinterface 1616. Traveling waves may be measured and recorded inreal-time, since they are transient signals that dissipate rapidly in anelectric power delivery system. Data acquisition module 1640 may operatein conjunction with fault detector module 1634. Data acquisition module1640 may control recording of data used by the fault detector module1634. According to one embodiment, data acquisition module 1640 mayselectively store and retrieve data and may make the data available forfurther processing. Such processing may include processing by faultdetector module 1634, which may be configured to determine theoccurrence of a fault with an electric power distribution system.

An incremental quantities module 1636 may be configured to calculatetime domain incremental quantities based on the techniques disclosedherein. The incremental quantities module 1636 may be configured to usedigitized representations of current and/or voltage measurements tocalculate incremental quantities therefrom. In some embodiments, system1600 may be one of a pair of IEDs in communication with differentterminals on an electric power system such as the IEDs and system ofFIG. 1. In one embodiment, each IED of a pair of IEDs calculatesincremental quantities in its own incremental quantities module 1636 forlater processing and sharing between the IEDs. In another embodiment,system 1600 may receive digitized representations from both the sensorcomponent 1610 and from a remote IED over a communications channel, andthe incremental quantities module 1636 may be configured to calculateincremental signals from both sources to calculate both local and remoteincremental quantities.

A fault type module 1638 may be configured to determine a fault typeusing incremental quantities from module 1636. Fault type module 1638may use the techniques disclosed herein, including equations from theoperating quantities listed in Table 2 to determine a fault type andprovide the proper incremental quantities to use for other processingwithin the IED.

Traveling wave differential module 1644 may determine a controloperation to take due to occurrence of a fault by determining adirection to a fault using current quantities only in traveling wavedifferential calculations. Traveling wave differential module 1644 mayuse equations 38-44. Traveling wave differential module 1644 may operateaccording to the method illustrated in FIG. 15.

A correlation module 1648 may be configured to receive local and remoteincremental quantities, and to correlate them. The correlation may bedone by time alignment using time stamps.

A directional module 1650 may be configured to determine a direction(forward or reverse) to a fault. The directional module 1650 may beconfigured to use incremental quantities from incremental quantitiesmodule 1636 to determine a direction to a fault. The directional module1650 may use equations 13-21. In various embodiments, directional module1650 may operate according to FIG. 10. In other embodiments, directionalmodule 1650 may be configured to determine the direction based on thepolarity of traveling waves. In such embodiments, the polarities of thevoltage and current traveling waves are opposite if the fault is in theforward direction. If the fault is in the reverse direction, the voltageand current traveling waves have same polarity.

A protective action module 1652 may be configured to implement aprotective action based on the declaration of a fault by the faultdetector module 1634. In various embodiments, a protective action mayinclude tripping a breaker, selectively isolating a portion of theelectric power system, etc. In various embodiments, the protectiveaction module 1652 may coordinate protective actions with other devicesin communication with system 1600.

In various embodiments system 1600 may be configured to provideprotection based on instantaneous voltages and currents. Such signalcomponents require shorter data windows but facilitate fasterprotection. Various embodiments of system 1600 may be configured toachieve an operating time of approximately 1 millisecond. Such a systemmay utilize a lumped parameter circuit-based and TW-based time-domainapproach and may allow for versatile applications covering various relayinput voltage sources and available communications channels. Such asystem may utilize high sampling rates (≧1 MHz), high-resolution (≧16bits) synchronized sampling, high-fidelity time synchronization, and acommunications network capable of exchanging all acquired data (≧100Mbps), or high numeric burden required by some of the algorithms (≧1 Gmultiplications per second).

Although several embodiments discussed hereinabove refer to three phasesof an alternating-current electric power delivery system, the principlesherein may be applied to a multiple-phase alternating-current electricpower system having more or less than three phases. For example, afour-phase electric power delivery system is contemplated, as is asix-phase electric power delivery system. The principles taught hereinmay be applied. In other embodiments, the principles taught may beapplied to a direct-current electric power delivery system. Inparticular, traveling wave detection using currents only in a travelingwave differential module may use current quantities from adirect-current electric power delivery system to detect faults and takecontrol actions thereon.

While specific embodiments and applications of the disclosure have beenillustrated and described, it is to be understood that the disclosure isnot limited to the precise configuration and components disclosedherein. Various modifications, changes, and variations apparent to thoseof skill in the art may be made in the arrangement, operation, anddetails of the methods and systems of the disclosure without departingfrom the spirit and scope of the disclosure.

What is claimed is:
 1. A system configured to detect a fault in anelectric power delivery system, comprising: a data acquisition subsystemconfigured to receive a plurality of representations of electricalconditions associated with at least a portion of the electric powerdelivery system; a traveling wave differential subsystem configured to:determine an operating quantity based on the plurality ofrepresentations of electrical conditions associated with the at least aportion of the electric power delivery system; determine a restraintquantity based on the plurality of representations of electricalconditions associated with the at least a portion of the electric powerdelivery system; detect a traveling wave generated by the fault based onthe plurality of representations of electrical conditions associatedwith the at least a portion of the electric power delivery system; adirectional subsystem configured to: determine a direction of the faultbased on a polarity of the traveling wave; and determine that the faultfalls within a zone of protection based on the direction of the fault; afault detector subsystem configured to declare a fault based on acomparison of the operating quantity and the restraint quantity; and aprotective action subsystem configured to implement a protective actionbased on the declaration of the fault.
 2. The system of claim 1, furthercomprising: a correlation subsystem configured to correlate theplurality of representations based on an indication associating eachrepresentation with one of a first end of a transmission line and asecond end of a transmission line.
 3. The system of claim 2, wherein thecorrelation subsystem is further configured to correlate the pluralityof representations based on a corresponding plurality of time stamps. 4.The system of claim 3, further comprising a communication interfaceconfigured to receive at least a subset of the plurality ofrepresentations of electrical conditions and the corresponding pluralityof time stamps from a remote device.
 5. The system of claim 1, whereinthe traveling wave differential subsystem is further configured todetermine the operating quantity based a summation of a subset of theplurality of representations associated with the traveling wave at afirst end of the electric power delivery system and a second end of theelectric power delivery system.
 6. The system of claim 5, wherein thetraveling wave differential subsystem is further configured to determinethe summation of the subset of the plurality of representationsassociated with the traveling wave based on a line propagation delayapplied to the traveling wave at each of the first end and the secondend of the electric power delivery system.
 7. The system of claim 1,wherein the traveling wave differential subsystem is further configuredto determine the restraint quantity based on a determination of apassing through traveling wave traversing from a first end of theelectric power delivery system to a second end of the electric powerdelivery system.
 8. The system of claim 1, wherein the traveling wavedifferential subsystem is configured to process the plurality ofrepresentations of electrical conditions at a rate of from around500,000 representations per second to around 5 million representationsper second.
 9. The system of claim 8, wherein the traveling wavedifferential subsystem is configured to process the plurality ofrepresentations of electrical conditions at a rate of about 1 millionrepresentations per second.
 10. The system of claim 1, wherein the zoneof protection comprises a reach setting configured to specify a distancebeyond which implementation of the protective action is inhibited. 11.The system of claim 1, wherein the directional subsystem is furtherconfigured to determine that the direction by comparing the polaritiesof an incremental voltage and an incremental current, and determiningthat the fault is in the forward direction when the incremental voltageand incremental current are of the same polarity.
 12. The system ofclaim 1, wherein the plurality of representations of electricalconditions represent time-domain quantities.
 13. The system of claim 1,wherein the plurality of representations of electrical conditionsrepresent only measurements of electrical currents.
 14. A method fordetecting a fault in an electric power delivery system, comprising:receiving a plurality of representations of electrical conditionsassociated with at least a portion of the electric power deliverysystem; determining an operating quantity based on the plurality ofrepresentations of electrical conditions associated with the at least aportion of the electric power delivery system; determining a restraintquantity based on the plurality of representations of electricalconditions associated with the at least a portion of the electric powerdelivery system; detecting a traveling wave generated by the fault basedon the plurality of representations of electrical conditions associatedwith the at least a portion of the electric power delivery system;determining a direction of the fault based on a polarity of thetraveling wave; determining based on the direction of the fault that thefault falls within a zone of protection; declaring a fault based on acomparison of the operating quantity and the restraint quantity; andimplementing a protective action based on the declaration of the fault.15. The method of claim 14, further comprising correlating the pluralityof representations based on a corresponding indication associating eachrepresentation with one of a first end of a transmission line and asecond end of a transmission line.
 16. The method of claim 14, furthercomprising correlating the plurality of representations based on acorresponding plurality of time stamps.
 17. The method of claim 14,wherein determining the operating quantity further comprises determininga summation of a subset of the plurality of representations associatedwith the traveling wave at a first end of the electric power deliverysystem and a second end of the electric power delivery system.
 18. Themethod of claim 17, wherein determining the summation of the subset ofthe plurality of representations associated with the traveling wavefurther comprises adjusting for a line propagation delay between each ofthe first end and the second end of the electric power delivery system.19. The method of claim 15, wherein determining the restraint quantitycomprises determining a passing through traveling wave traversing from afirst end of the electric power delivery system to a second end of theelectric power delivery system.
 20. The method of claim 14, furthercomprising determining the zone of protection based on a reach settingconfigured to specify a distance beyond which implementation of theprotective action is inhibited.
 21. The method of claim of claim 14,wherein determining the direction to the fault comprises: comparing thepolarities of the incremental voltage and the incremental current, anddetermining that the fault is in the forward direction when theincremental voltage and incremental current are of the same polarity.22. The method of claim 14, wherein the plurality of representations ofelectrical conditions represent only measurements of electricalcurrents.
 23. A system configured to detect a fault in an electric powerdelivery system, comprising: a data acquisition subsystem configured toreceive a plurality of representations of electrical conditionsassociated with at least a portion of the electric power deliverysystem; a traveling wave differential subsystem configured to: determinean operating quantity based on the plurality of representations ofelectrical conditions associated with the at least a portion of theelectric power delivery system; determine a restraint quantity based onthe plurality of representations of electrical conditions associatedwith the at least a portion of the electric power delivery system bydetermining a passing through traveling wave traversing from a first endof the electric power delivery system to a second end of the electricpower delivery system; detect a traveling wave generated by the faultbased on the plurality of representations of electrical conditionsassociated with the at least a portion of the electric power deliverysystem; a fault detector subsystem configured to declare a fault basedon a comparison of the operating quantity and the restraint quantity;and a protective action subsystem configured to implement a protectiveaction based on the declaration of the fault.
 24. A method for detectinga fault in an electric power delivery system, comprising: receiving aplurality of representations of electrical conditions associated with atleast a portion of the electric power delivery system; determining anoperating quantity based on the plurality of representations ofelectrical conditions associated with the at least a portion of theelectric power delivery system; determining a restraint quantity basedon the plurality of representations of electrical conditions associatedwith the at least a portion of the electric power delivery system bydetermining a passing through traveling wave traversing from a first endof the electric power delivery system to a second end of the electricpower delivery system; detecting a traveling wave generated by the faultbased on the plurality of representations of electrical conditionsassociated with the at least a portion of the electric power deliverysystem; declaring a fault based on a comparison of the operatingquantity and the restraint quantity; and implementing a protectiveaction based on the declaration of the fault.